The Heston stochastic volatility model has a boundary trace at zero volatility

Abstract

We establish boundary regularity results in Hölder spaces for the degenerate parabolic problem obtained from the Heston stochastic volatility model in Mathematical Finance set up in the spatial domain (upper half-plane) mathbb {H} = mathbb {R}times (0,infty )subset mathbb {R}^2. Starting with nonsmooth initial data u_0in H, we take advantage of smoothing properties of the parabolic semigroup textrm{e}^{-t{mathcal {A}}}:,Hrightarrow H, tin mathbb {R}_+, generated by the Heston model, to derive the smoothness of the solution u(t) = textrm{e}^{-t{mathcal {A}}} u_0 for all t>0. The existence and uniqueness of a weak solution is obtained in a Hilbert space H = L^2(mathbb {H};mathfrak {w}) with very weak growth restrictions at infinity and on the boundary partial mathbb {H}= mathbb {R}times { 0}subset mathbb {R}^2 of the half-plane mathbb {H}. We investigate the influence of the boundary behavior of the initial data u_0in H on the boundary behavior of u(t) for t>0.